The High-d Limit: General Properties and Weak-Coupling Perturbation Theory

General Remarks

It is a well-known fact that theoretical investigations of
quantum-mechanical many-body systems are faced with severe technical
problems - particularly in those dimensions which are most interesting
to us, i. e. d=2,3. This is due to the complicated dynamics of
such systems and the non-trivial algebra needed to describe the
quantum-mechanical world. In the absence of exact methods there is
clearly a great need for reliable, controlled approximation
schemes. The construction of such schemes is not at all
straightforward. In fact, for fermionic lattice models even the
familiar concept of a "mean field theory" is considerably more
delicate than in the case of most classical systems. There exists a
well-established branch of approximation techniques which makes use of
the simplifications that occur when some parameter (e.g. the length of
the spins S, the spin degeneracy N, etc.) is taken to
be large (in fact, infinite). Investigations in this limit,
supplemented by an expansion in the inverse of the large parameter,
may even provide valuable insight into the fundamental properties of a
system where this parameter is not large.

In this set of lectures I will discuss a new approach to fermionic
systems, based on the limit of high spatial dimensions d. The
limit d→∞, which is
well-known from classical physics (e. g. the Ising model), is not as
academic as it might seem. In fact, we will find that in many respects
the dimension d=3 of our real world is already high.
In particular, we will discover that a large number of standard
approximations, which are commonly used to explain experimental
results, are only correct in d=∞. In this respect the limit d→∞ is useful even for
physicists who are mainly interested in the application of theories.

Motivation for the large-dimension limit

In a perfectly crystalline system every lattice site has the same
number of nearest neighbors (coordination number Z). In three
dimensions (d=3) one has Z=6 for a simple cubic lattice
(Z=2d for a hypercubic lattice in general dimensions d),
Z=8 for a bcc lattice and Z=12 for an fcc-lattice. The
dimensionality of a lattice system is directly described by the number
Z, rather than by the somewhat more abstract "number of
dimensions d". Since Z~O(10) is already quite
large in d=3, such that 1/Z is rather small, it is only
natural and in the general spirit of theoretical physics to consider
the extreme limit Z→∞
first, and then use 1/Z as a small expansion parameter to reach
finite Z.